Optimal Convergence Rate of the Randomized Algorithms of Stochastic Approximation in Arbitrary Noise

Multidimensional stochastic optimization plays an important role in analysis and control of many technical systems. To solve the challenging multidimensional problems of optimization, it was suggested to use the randomized algorithms of stochastic approximation with perturbed input which are not just simple, but also provide consistent estimates of the unknown parameters for observations in almost arbitrary noise. Optimal methods of choosing the parameters of algorithms were motivated.     

Chunky and equal-spaced polynomial multiplication

Finding the product of two polynomials is an essential and basic problem in computer algebra. While most previous results have focused on the worst-case complexity, we instead employ the technique of adaptive analysis to give an improvement in many “easy” cases. We present two adaptive measures and methods for polynomial multiplication, and also show how to effectively combine them to gain both advantages. One useful feature of these algorithms is that they essentially provide a gradient between existing “sparse” and “dense” methods. We prove that these approaches provide significant improvements in many cases but in the worst case are still comparable to the fastest existing algorithms.     

Genetic algorithms to solve the cover printing problem

Inspired by successful application of evolutionary algorithms to solving difficult optimization problems, we explore in this paper, the applicability of genetic algorithms (GAs) to the cover printing problem, which consists in the grouping of book covers on offset plates in order to minimize the total production cost. We combine GAs with a linear programming solver and we propose some innovative features such as the “unfixed two-point crossover operator” and the “binary stochastic sampling with replacement” for selection. Two approaches are proposed: an adapted genetic algorithm and a multiobjective genetic algorithm using the Pareto fitness genetic algorithm. The resulting solutions are compared. Some computational experiments have also been done to analyze the effects of different genetic operators on both algorithms.     

Parallel evolutionary algorithms can achieve super-linear performance

One of the main reasons for using parallel evolutionary algorithms (PEAs) is to obtain efficient algorithms with an execution time much lower than that of their sequential counterparts in order, e.g., to tackle more complex problems. This naturally leads to measuring the speedup of the PEA. PEAs have sometimes been reported to provide super-linear performances for different problems, parameterizations, and machines. Super-linear speedup means that using “m” processors leads to an algorithm that runs more than “m” times faster than the sequential version. However, reporting super-linear speedup is controversial, especially for the “traditional” research community, since some non-orthodox practices could be thought of being the cause for this result. Therefore, we begin by offering a taxonomy for speedup, in order to clarify what is being measured. Also, we analyze the sources for such a scenario in this paper. Finally, we study an assorted set of results. Our conclusion is that super-linear performance is possible for PEAs, theoretically and in practice, both in homogeneous and in heterogeneous parallel machines.     

Priority algorithms for graph optimization problems

We continue the study of priority or “greedy-like” algorithms as initiated in Borodin et al. (2003) [10] and as extended to graph theoretic problems in Davis and Impagliazzo (2009) [12]. Graph theoretic problems pose some modeling problems that did not exist in the original applications of Borodin et al. and Angelopoulos and Borodin (2002) [3]. Following the work of Davis and Impagliazzo, we further clarify these concepts. In the graph theoretic setting, there are several natural input formulations for a given problem and we show that priority algorithm bounds in general depend on the input formulation. We study a variety of graph problems in the context of arbitrary and restricted priority models corresponding to known “greedy algorithms”.     

Improved artificial bee colony algorithm for global optimization

The artificial bee colony algorithm is a relatively new optimization technique. This paper presents an improved artificial bee colony (IABC) algorithm for global optimization. Inspired by differential evolution (DE) and introducing a parameter M, we propose two improved solution search equations, namely “ABC/best/1” and “ABC/rand/1”. Then, in order to take advantage of them and avoid the shortages of them, we use a selective probability p to control the frequency of introducing “ABC/rand/1” and “ABC/best/1” and get a new search mechanism. In addition, to enhance the global convergence speed, when producing the initial population, both the chaotic systems and the opposition-based learning method are employed. Experiments are conducted on a suite of unimodal/multimodal benchmark functions. The results demonstrate the good performance of the IABC algorithm in solving complex numerical optimization problems when compared with thirteen recent algorithms.     

Disturbed Exploitation compact Differential Evolution for limited memory optimization problems

This paper proposes a novel and unconventional Memetic Computing approach for solving continuous optimization problems characterized by memory limitations. The proposed algorithm, unlike employing an explorative evolutionary framework and a set of local search algorithms, employs multiple exploitative search within the main framework and performs a multiple step global search by means of a randomized perturbation of the virtual population corresponding to a periodical randomization of the search for the exploitative operators. The proposed Memetic Computing approach is based on a populationless (compact) evolutionary framework which, instead of processing a population of solutions, handles its statistical model. This evolutionary framework is based on a Differential Evolution which cooperatively employs two exploitative search operators: the first is based on a standard Differential Evolution mutation and exponential crossover, and the second is the trigonometric mutation. These two search operators have an exploitative action on the algorithmic framework and thus contribute to the rapid convergence of the virtual population towards promising candidate solutions. The action of these search operators is counterbalanced by a periodical stochastic perturbation of the virtual population, which has the role of “disturbing” the excessively exploitative action of the framework and thus inhibits its premature convergence. The proposed algorithm, namely Disturbed Exploitation compact Differential Evolution, is a simple and memory-wise cheap structure that makes use of the Memetic Computing paradigm in order to solve complex optimization problems. The proposed approach has been tested on a set of various test problems and compared with state-of-the-art compact algorithms and with some modern population based meta-heuristics. Numerical results show that Disturbed Exploitation compact Differential Evolution significantly outperforms all the other compact algorithms present in literature and reaches a competitive performance with respect to modern population algorithms, including some memetic approaches and complex modern Differential Evolution based algorithms. In order to show the potential of the proposed approach in real-world applications, Disturbed Exploitation compact Differential Evolution has been implemented for performing the control of a space robot by simulating the implementation within the robot micro-controller. Numerical results show the superiority of the proposed algorithm with respect to other modern compact algorithms present in literature.     

Complexity reduction in MPC for stochastic max-plus-linear discrete event systems by variability expansion

Model predictive control (MPC) is a popular controller design technique in the process industry. Recently, MPC has been extended to a class of discrete event systems that can be described by a model that is “linear” in the max-plus algebra. In this context both the perturbations-free case and for the case with noise and/or modeling errors in a bounded or stochastic setting have been considered. In each of these cases an optimization problem has to be solved on-line at each event step in order to determine the MPC input. This paper considers a method to reduce the computational complexity of this optimization problem, based on variability expansion. In particular, it is shown that the computational load is reduced if one decreases the level of “randomness” in the system.     

Research on probabilistic methods for control system design

A novel approach based on probability and randomization has emerged to synergize with the standard deterministic methods for control of systems with uncertainty. The main objective of this paper is to provide a broad perspective on this area of research known as “probabilistic robust control”, and to address in a systematic manner recent advances. The focal point is on design methods, based on the interplay between uncertainty randomization and convex optimization, and on the illustration of specific control applications.     

Randomized algorithms for probabilistic robustness with real andcomplex structured uncertainty

There has been a growing interest in developing randomized algorithms for probabilistic robustness of uncertain control systems. Unlike classical worst case methods, these algorithms provide probabilistic estimates assessing, for instance, if a certain design specification is met with a given probability. One of the advantages of this approach is that the robustness margins can be often increased by a considerable amount, at the expense of a small risk. In this sense, randomized algorithms may be used by the control engineer together with standard worst case methods to obtain additional useful information. The applicability of these probabilistic methods to robust control is presently limited by the fact that the sample generation is feasible only in very special cases which include systems affected by real parametric uncertainty bounded in rectangles or spheres. Sampling in more general uncertainty sets is generally performed through overbounding, at the expense of an exponential rejection rate. In the paper, randomized algorithms for stability and performance of linear time invariant uncertain systems described by a general M-Δ configuration are studied. In particular, efficient polynomial-time algorithms for uncertainty structures Δ consisting of an arbitrary number of full complex blocks and uncertain parameters are developed     

Probabilistic reachability and safety for controlled discrete time stochastic hybrid systems

In this work, probabilistic reachability over a finite horizon is investigated for a class of discrete time stochastic hybrid systems with control inputs. A suitable embedding of the reachability problem in a stochastic control framework reveals that it is amenable to two complementary interpretations, leading to dual algorithms for reachability computations. In particular, the set of initial conditions providing a certain probabilistic guarantee that the system will keep evolving within a desired ‘safe’ region of the state space is characterized in terms of a value function, and ‘maximally safe’ Markov policies are determined via dynamic programming. These results are of interest not only for safety analysis and design, but also for solving those regulation and stabilization problems that can be reinterpreted as safety problems. The temperature regulation problem presented in the paper as a case study is one such case.     

On the determination of optimal costly measurement strategies for linear stochastic systems

This paper presents the formulation of a class of optimization problems dealing with selecting, at each instant of time, one measurement provided by one out of many sensors. Each measurement has an associated measurement cost. The basic problem is then to select an optimal measurement policy, during a specified observation time interval, so that a weighted combination of “prediction accuracy” and accumulated “observation cost” is optimized. The current analysis is limited to the class of linear stochastic dynamic systems and measurement subsystems. The problem of selecting the optimal measurement strategy can be transformed into a deterministic optimal control problem. An iterative digital computer algorithm is suggested for obtaining numerical results. It is shown that the optimal measurement policy and the associated “matched” Kalman-type filter can be precomputed, i.e. specified before the measurements actually occur. Numerical results for a third-order system with two possible measurements are presented.     

An improved GA and a novel PSO-GA-based hybrid algorithm

Inspired by the natural features of the variable size of the population, we present a variable population-size genetic algorithm (VPGA) by introducing the “dying probability” for the individuals and the “war/disease process” for the population. Based on the VPGA and the particle swarm optimization (PSO) algorithms, a novel PSO-GA-based hybrid algorithm (PGHA) is also proposed in this paper. Simulation results show that both VPGA and PGHA are effective for the optimization problems.     

(Incremental) Priority Algorithms

We study the question of which optimization problems can be optimally or approximately solved by greedy or greedy-like algorithms. For definiteness, we limit the present discussion to some well-studied scheduling problems although the underlying issues apply in a much more general setting. Of course, the main benefit of greedy algorithms lies in both their conceptual simplicity and their computational efficiency. Based on the experience from online competitive analysis, it seems plausible that we should be able to derive approximation bounds for greedy-like algorithms exploiting only the conceptual simplicity of these algorithms. To this end, we need (and will provide) a precise definition of what we mean by greedy and greedy-like.     

An investigation of mating and population maintenance strategies in hybrid genetic heuristics for product line designs

This research builds on prior work on developing near optimal solutions to the product line design problems within the conjoint analysis framework. In this research, we investigate and compare different genetic algorithm operators; in particular, we examine systematically the impact of employing alternative population maintenance strategies and mutation techniques within our problem context. Two alternative population maintenance methods, that we term “Emigration” and “Malthusian” strategies, are deployed to govern how individual product lines in one generation are carried over to the next generation. We also allow for two different types of reproduction methods termed “Equal Opportunity” in which the parents to be paired for mating are selected with equal opportunity and a second based on always choosing the best string in the current generation as one of the parents which is referred to as the “Queen bee”, while the other parent is randomly selected from the set of parent strings. We also look at the impact of integrating the artificial intelligence approach with a traditional optimization approach by seeding the GA with solutions obtained from a Dynamic Programming heuristic proposed by others. A detailed statistical analysis is also carried out to determine the impact of various problem and technique aspects on multiple measures of performance through means of a Monte Carlo simulation study. Our results indicate that such proposed procedures are able to provide multiple “good” solutions. This provides more flexibility for the decision makers as they now have the opportunity to select from a number of very good product lines. The results obtained using our approaches are encouraging, with statistically significant improvements averaging 5% or more, when compared to the traditional benchmark of the heuristic dynamic programming technique.     

A mathematical model on EPQ for stochastic demand in an imperfect production system

In the paper, we develop an EPQ (economic production quantity) inventory model to determine the optimal buffer inventory for stochastic demand in the market during preventive maintenance or repair of a manufacturing facility with an EPQ (economic production quantity) model in an imperfect production system. Preventive maintenance, an essential element of the just-in-time structure, may cause shortage which is reduced by buffer inventory. The products are sold with the free minimal repair warranty (FRW) policy. The production system may undergo “out-of-control” state from “in-control” state, after a certain time that follows a probability density function. The defective (non-conforming) items in “in-control” or “out-of-control” state are reworked at a cost just after the regular production time. Finally, an expected cost function regarding the inventory cost, unit production cost, preventive maintenance cost and shortage cost is minimized analytically. We develop another case where the buffer inventory as well as the production rate are decision variables and the expected unit cost considering the above cost functions is optimized also. The numerical examples are provided to illustrate the behaviour and application of the model. Sensitivity analysis of the model with respect to key parameters of the system is carried out.     

On the complexity of working set selection

The decomposition method is currently one of the major methods for solving the convex quadratic optimization problems being associated with Support Vector Machines (SVM-optimization). A key issue in this approach is the policy for working set selection. We would like to find policies that realize (as well as possible) three goals simultaneously: “(fast) convergence to an optimal solution”, “efficient procedures for working set selection”, and “a high degree of generality” (including typical variants of SVM-optimization as special cases). In this paper, we study a general policy for working set selection that has been proposed in [Nikolas List, Hans Ulrich Simon, A general convergence theorem for the decomposition method, in: Proceedings of the 17th Annual Conference on Computational Learning Theory, 2004, pp. 363–377] and further analyzed in [Nikolas List, Hans Ulrich Simon, General polynomial time decomposition algorithms, in: Proceedings of the 17th Annual Conference on Computational Learning Theory, 2005, pp. 308–322]. It is known that it efficiently approaches feasible solutions with minimum cost for any convex quadratic optimization problem. Here, we investigate its computational complexity when it is used for SVM-optimization. It turns out that, for a variable size of the working set, the general policy poses an NP-hard working set selection problem. But a slight variation of it (sharing the convergence properties with the original policy) can be solved in polynomial time. For working sets of fixed size 2, the situation is even better. In this case, the general policy coincides with the “rate certifying pair approach” (introduced by Hush and Scovel). We show that maximum rate certifying pairs can be found in linear time, which leads to a quite efficient decomposition method with a polynomial convergence rate for SVM-optimization.     

Comparison of simple diversity mechanisms on plateau functions

It is widely assumed and observed in experiments that the use of diversity mechanisms in evolutionary algorithms may have a great impact on its running time. Up to now there is no rigorous analysis pointing out how different diversity mechanisms influence the runtime behavior. We consider evolutionary algorithms that differ from each other in the way they ensure diversity and point out situations where the right mechanism is crucial for the success of the algorithm. The considered evolutionary algorithms either diversify the population with respect to the search points or with respect to function values. Investigating simple plateau functions, we show that using the “right” diversity strategy makes the difference between an exponential and a polynomial runtime. Later on, we examine how the drawback of the “wrong” diversity mechanism can be compensated by increasing the population size.     

Learning with Genetic Algorithms: An Overview

Genetic algorithms represent a class of adaptive search techniques that have been intensively studied in recent years. Much of the interest in genetic algorithms is due to the fact that they provide a set of efficient domain-independent search heuristics which are a significant improvement over traditional weak methods without the need for incorporating highly domain-specific knowledge. There is now considerable evidence that genetic algorithms are useful for global function optimization and NP-hard problems. Recently, there has been a good deal of interest in using genetic algorithms for machine learning problems. This paper provides a brief overview of how one might use genetic algorithms as a key element in learning systems.     

Dwell-time controllers for stochastic systems with switching Markov chain

We study the problem of feedback stabilization of a family of nonlinear stochastic systems with switching mechanism modeled by a Markov chain. We introduce a novel notion of stability under switching, which guarantees a given probability that the trajectories of the system hit some target set in finite time and remain thereinafter. Our main contribution is to prove that if the expectation of the time between two consecutive switching (dwell time) is “sufficiently large”, then the system is stable under switching with guaranteed probability. We illustrate this methodology by constructing measurement feedback controllers for a wide class of stochastic nonlinear systems.